Trade-offs Between Mobility and Density for Coverage in

Wireless Sensor Networks

Wei Wang, Vikram Srinivasan and Kee-Chaing Chua

National University of Singapore

2007 Mobicom

Outline

Introduction Coverage with mobile sensors Coverage of hybrid networks Mobility algorithm Numerical results Conclusion

Introduction

Coverage problem Important research problem in WSNs k-covered Network Deployment Mobility

Introduction- deployment

Metric: over-provisioning factor Indicates the efficiency of a network deployment s

trategy Consider a random deployment strategy

What is the sensor density to guarantee k-coverage?

Introduction- mobility

Mobile sensors can relocate themselves to heal coverage holes Over-provisioning factor for a network with all

mobile sensors can be Θ(1) Consumes more energy

Mobile sensors Limited mobility: move once, over a short distance

Maximum distance?

Coverage with mobile sensors

Sensing field: L=l*l Num. of static sensors: N = λL

Uniformly and independently scattered in the network. Number of static sensors in a region with area of A:

nA

Sensing range: r = 1 /√π 1=πr2 1

Density

Over-Provisioning Factor

Optimal over-provisioning factor:Θ(1) ds= √2r

Density of mobile sensor

K-coverage

r = 1 /√π

Over-Provisioning Factor

Randomly deployed static sensor networks Density λ

Total expected area which is uncovered is e−λL. Random coverage processes Large enough λ, e−λ can be made arbitrarily small

Probability approaches one for a network with constant sensor density λ when the network size L→∞. Exist a connected coverage hole larger than unit area

Over-Provisioning Factor

To achieve k-coverage in a large network, the static sensor density needs to grow with the network size λ = logL +(k + 2) log log L + c(L)

c(L) → +∞ as L → +∞

All Mobile Networks

ηm = Θ(1). key question

what is the maximum distance that each sensor has to move?

Limit the maximum moving distance for each mobile

All Mobile Networks

Theorem1: Network can provide k-coverage with an over-provisioning factor of ηm= π/ 2 and the maximum distancemaximum distance moved by any mobile sensor is O( 1 √klog3/4(kL)) w.h.p.

All Mobile Networks

Sensing field into square grids with side length of da =√2r/√k Number of nodes in the sensing range

πr2/(√2r/√k)2=πk/2

ηm=(πk/2) / k = π/2

All Mobile Networks

By the lower bounds on lattice points covered by a circle, there are at least W(k) lattice points of side length of da covered by a circle of radius r

da =√2r/√k Increasing function

All Mobile Networks

W(k) > k when k ≥ 25 ->k coverage W(k)=25.13274

Network is at least k-covered when 1 ≤ k < 25.

All Mobile Networks

l × l square, L = l2 points in the region there exists a perfect match between the L rando

m points and the L grid points with maximum distance between any matched pairs of O(log3/4 L).

Grid points (k/2r2)*L O(log3/4 (kL))

Grid size is da =√2r /√k O( 1/√k log3/4(kL))

1=πr2

1/r2＝ πηm =Densty/kDensty= ηm*k= πk/2=k/2r2

Coverage of hybrid networks

Over-provisioning factor is O(1) Fraction of mobile sensors required is less th

an 1 /√2πk Maximum distance that any mobile sensor wil

l have to move is O(log3/4L)

Density of Mobile Sensors

Static sensor density at λ =2πk. Divide the network into square cells

equal side length of dh = r/√2.

Average number of static sensors in each cell will be 2πkd2

h = k.

Density of Mobile Sensors

The network will be k-covered if all cells contain at least k sensors. cell i has vi = k−ni vacancies, If a cell i contains ni

< k static sensors

Poisson approximation

Density of Mobile Sensors

The random variable vi = [k − ni]+ , will be distributed as:

The expected number of vacancies in a cell will be:

Density of Mobile Sensors

Using Stirling’s approximation

Density of mobile sensor

Density of Static sensor

Fraction of mobile sensors required is less than

r = 1 /√πdh = r/√2.

Maximum distance for mobiles

A grid with side length of 1/ √Λ Maximum distance

Decreasing function Matching distance

Mobility Algorithm

Problem Formulation Movement cost

Initial number of mobile sensor

Number of mobile sensor from cell i to cell j

Distribution Solution

A distributed algorithm Maximum flow problem

Assume Sensor knows

Its location Which cell it is located in. vi and mi

Each cell elects a mobile or static sensor as the delegate Communicate and exchange information with its neighbors in

graph G

Distribution Solution-push-relabel algorithm

a

b c

i o

o i o i

Cell a

Cell a Cell c

Distance D v-m=3

v-m=-2 v-m=-1

Distribution Solution-push-relabel algorithm a

b c

i o

o i o i

Cell a

Cell a Cell c

h(i)=0e(i)=0

h(i) =0e(i) =0

h(i) =0e(i) =0

h(o)=0e(o)=3

h(o) =0e(o) =-2

h(o)=0e(o) =-1

Zero cost

ci

v-m=3

v-m=-2 v-m=-1

Distribution Solution-push-relabel algorithm a

b c

i o

o i o i

Cell a

Cell a Cell c

h(i)=0e(i)=0

h(i) =0e(i) =0

h(i) =0e(i) =0

h(o)=0e(o)=3

h(o) =0e(o) =-2

h(o)=0e(o) =-1

v-m=3

v-m=-2 v-m=-1h(o)=1e(o)=3

Distribution Solution-push-relabel algorithm a

b c

i o

o i o i

Cell a

Cell a Cell c

h(i)=0e(i)=0

h(i) =0e(i) =0

h(i) =0e(i) =1

h(o) =0e(o) =-2

h(o)=0e(o) =-1

v-m=3

v-m=-2 v-m=-1h(o)=1e(o)=2h(o)=1e(o)=1

h(i) =0e(i) =1

Distribution Solution-push-relabel algorithm a

b c

i o

o i o i

Cell a

Cell a Cell c

h(i)=0e(i)=0

h(i) =0e(i) =0

h(o) =0e(o) =-1

h(o)=0e(o) =1

v-m=3

v-m=-2 v-m=-1h(o)=1e(o)=1

h(i) =0e(i) =0

Distribution Solution-push-relabel algorithm a

b c

i o

o i o i

Cell a

Cell a Cell c

h(i)=0e(i)=0

h(i) =0e(i) =0

h(o) =0e(o) =-1

h(o)=0e(o) =1

v-m=3

v-m=-2 v-m=-1h(o)=1e(o)=1

h(i) =0e(i) =1

Numerical results

Mobile Sensor Networks only consider the maximum matching distance for

1-coverage in our simulations M = ΛL mobiles

Λ=π/2 ds= √2 r 105 randomly generated topologies

Probability that no feasible matching exists for a given maximum moving distance D.

ds

Numerical results

Hybrid Networks Cells with side length of dh = r/√2 N = λL static sensors , λ = 2πk M = ΛL mobiles

M is selected so that there are exactly enough mobiles to fill all vacancies

Moving distance D

k=10

dh=0.5 ds

Cells=900

Performance of Push-Relabel Algorithm

Execution process is divided into rounds 103 randomly generated topologies

Total number of messagesRounds

Conclusion

Investigate the distance that a mobile sensor will have to move Mobile sensor networks Hybrid sensor networks

Results prove that Mobility has significant advantages in providing

coverage